A description/proof of that area of hyperrectangle can be approximated by area of covering finite number hypersquares to any precision
Topics
About: Euclidean metric space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that the area of any hyperrectangle can be approximated by the area of covering finite number hypersquares to any precision.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
The area, \(a\), of any hyperrectangle on any \(\mathbb{R}^n\) Euclidean metric space can be approximated by the area of some finite number of hypersquares, \(a_i\), that cover the hyperrectangle, to any precision, which is, for any real number \(\epsilon \gt 0\), \(\sum _i a_i - a \lt \epsilon\).
2: Proof
Denote the side lengths of the hyperrectangle as \(l_1, l_2, . . ., l_n\). For any real number, \(l_s \gt 0\), and each side index, \(1 \leq i \leq n\), there is the non-negative unique integer, \(k_i (l_s)\), and the unique real number, \(0 \leq l'_i (l_s) \lt l_s\), such that \(k_i (l_s) l_s = l'_i (l_s) + l_i\), which means that the \(l_i\) length is covered by \(k_i (l_s)\) times \(l_s\) with the excess, \(l'_i (l_s)\), where the unique existences are clear based on the meaning.
Now, \(l_s\) is the side length of the same-size hypersquares that cover the hyperrectangle, starting from a vertex of the hyperrectangle, \(k_i (l_s)\) times for each \(i\) direction, with the excess of \(l'_i (l_s)\) for the \(i\) direction.
The excess area is the error of the approximation, which is \(e (l_s) = \prod _i l_s k_i (l_s) - \prod _i l_i\). But by directly calculating the excess area, \(e (l_s) \lt \sum _i l'_i (l_s) \prod _{j \neq i} (l'_j (l_s) + l_j)\), adding some areas multiple times. Assuming \(l_s \leq 1\) (which we are free to do) , \(\sum _i l'_i (l_s) \prod _{j \neq i} (l'_j (l_s) + l_j) \lt \sum _i l_s \prod _ {j \neq i} (l_s + l_j) \lt l_s \sum _i \prod _ {j \neq i} (1 + l_j)\).
As \(L:= \sum _i \prod _ {j \neq i} (1 + l_j)\) depends only the configuration of the hyperrectangle, we choose \(l_s\) as \(min (1, \frac{\epsilon}{L})\), then, \(e (l_s) \lt l_s L \leq \epsilon\).
